Let $y_1, dots,y_n$ be i.i.d. random variables from $Y$ with density $$p(y;theta)=theta(1-y)^{theta-1}, theta > 0, y in(0,1)$$
and $T=-ln(1-Y)sim Exp(theta)$ with $theta$ rate parameter.
The log-likelihood is $l(theta)=n ln(theta)+sumln(1-y)$ and the MLE is $hat theta=-frac{n}{sum ln(1-y)}$.
I need to prove the (weak) consistency of $hat theta$. A sufficient condition is
$$begin{cases} lim_{n to infty}mathbb{E}(hat theta)=theta \ lim_{n to infty}mathbb{Var}(hat theta)=0 end{cases}$$
Based on MLE invariance I could write $hat theta=frac{n}{sum t}$, so $$lim_{n to infty}mathbb{E}(hat theta)=lim_{n to infty}mathbb{E}left(frac{n}{sum t}right)=lim_{n to infty}left(nfrac{1}{nfrac{1}{theta}}right)=theta$$
and $$mathbb{Var}(hat theta)=mathbb{Var}left(frac{n}{sum t}right)=n^2mathbb{Var}left(frac{1}{summathbb{Var}(t)}right)=n^2frac{1}{frac{n}{theta^2}}=ntheta^2$$
What am I doing wrong?
Edit with solution
My solution: if $sum t_i sim Gamma(n,theta)$ then $frac{1}{sum t_i}sim IG(n,theta)$. Now, the variance of $IG$ should be $mathbb{Var}(IG(n,theta))=frac{theta^{2}}{(n-1)^2(n-2)}$ so $lim_{n to infty}n^2frac{theta^{2}}{(n-1)^2(n-2)}=0$ and this prove the consistency of $hat theta$.
Best Answer
The error you have made is that $$text{Var}left( dfrac{1}{sum t_i} right) ne dfrac{1}{sum text{Var}(t_i)}.$$
Instead you need to find the distribution of $sum t_i$, where $t_i sim Exp(theta)$. Then find the distribution of $1/sum t_i$. Hint, you should end up with an Inverse Gamma distribution. Find the variance of that distribution, and plug it in.